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Theorem imadifxp 23999
Description: Image of the difference with a cross product (Contributed by Thierry Arnoux, 13-Dec-2017.)
Assertion
Ref Expression
imadifxp  |-  ( C 
C_  A  ->  (
( R  \  ( A  X.  B ) )
" C )  =  ( ( R " C )  \  B
) )

Proof of Theorem imadifxp
StepHypRef Expression
1 ima0 5188 . . . 4  |-  ( ( R  \  ( A  X.  B ) )
" (/) )  =  (/)
2 imaeq2 5166 . . . 4  |-  ( C  =  (/)  ->  ( ( R  \  ( A  X.  B ) )
" C )  =  ( ( R  \ 
( A  X.  B
) ) " (/) ) )
3 imaeq2 5166 . . . . . . 7  |-  ( C  =  (/)  ->  ( R
" C )  =  ( R " (/) ) )
4 ima0 5188 . . . . . . 7  |-  ( R
" (/) )  =  (/)
53, 4syl6eq 2460 . . . . . 6  |-  ( C  =  (/)  ->  ( R
" C )  =  (/) )
65difeq1d 3432 . . . . 5  |-  ( C  =  (/)  ->  ( ( R " C ) 
\  B )  =  ( (/)  \  B ) )
7 0dif 3667 . . . . 5  |-  ( (/)  \  B )  =  (/)
86, 7syl6eq 2460 . . . 4  |-  ( C  =  (/)  ->  ( ( R " C ) 
\  B )  =  (/) )
91, 2, 83eqtr4a 2470 . . 3  |-  ( C  =  (/)  ->  ( ( R  \  ( A  X.  B ) )
" C )  =  ( ( R " C )  \  B
) )
109adantl 453 . 2  |-  ( ( C  C_  A  /\  C  =  (/) )  -> 
( ( R  \ 
( A  X.  B
) ) " C
)  =  ( ( R " C ) 
\  B ) )
11 inundif 3674 . . . . . . . . 9  |-  ( ( R  i^i  ( A  X.  B ) )  u.  ( R  \ 
( A  X.  B
) ) )  =  R
1211imaeq1i 5167 . . . . . . . 8  |-  ( ( ( R  i^i  ( A  X.  B ) )  u.  ( R  \ 
( A  X.  B
) ) ) " C )  =  ( R " C )
13 imaundir 5252 . . . . . . . 8  |-  ( ( ( R  i^i  ( A  X.  B ) )  u.  ( R  \ 
( A  X.  B
) ) ) " C )  =  ( ( ( R  i^i  ( A  X.  B
) ) " C
)  u.  ( ( R  \  ( A  X.  B ) )
" C ) )
1412, 13eqtr3i 2434 . . . . . . 7  |-  ( R
" C )  =  ( ( ( R  i^i  ( A  X.  B ) ) " C )  u.  (
( R  \  ( A  X.  B ) )
" C ) )
1514difeq1i 3429 . . . . . 6  |-  ( ( R " C ) 
\  B )  =  ( ( ( ( R  i^i  ( A  X.  B ) )
" C )  u.  ( ( R  \ 
( A  X.  B
) ) " C
) )  \  B
)
16 difundir 3562 . . . . . 6  |-  ( ( ( ( R  i^i  ( A  X.  B
) ) " C
)  u.  ( ( R  \  ( A  X.  B ) )
" C ) ) 
\  B )  =  ( ( ( ( R  i^i  ( A  X.  B ) )
" C )  \  B )  u.  (
( ( R  \ 
( A  X.  B
) ) " C
)  \  B )
)
1715, 16eqtri 2432 . . . . 5  |-  ( ( R " C ) 
\  B )  =  ( ( ( ( R  i^i  ( A  X.  B ) )
" C )  \  B )  u.  (
( ( R  \ 
( A  X.  B
) ) " C
)  \  B )
)
18 inss2 3530 . . . . . . . . 9  |-  ( R  i^i  ( A  X.  B ) )  C_  ( A  X.  B
)
19 imass1 5206 . . . . . . . . 9  |-  ( ( R  i^i  ( A  X.  B ) ) 
C_  ( A  X.  B )  ->  (
( R  i^i  ( A  X.  B ) )
" C )  C_  ( ( A  X.  B ) " C
) )
20 ssdif 3450 . . . . . . . . 9  |-  ( ( ( R  i^i  ( A  X.  B ) )
" C )  C_  ( ( A  X.  B ) " C
)  ->  ( (
( R  i^i  ( A  X.  B ) )
" C )  \  B )  C_  (
( ( A  X.  B ) " C
)  \  B )
)
2118, 19, 20mp2b 10 . . . . . . . 8  |-  ( ( ( R  i^i  ( A  X.  B ) )
" C )  \  B )  C_  (
( ( A  X.  B ) " C
)  \  B )
22 xpima 5280 . . . . . . . . . . 11  |-  ( ( A  X.  B )
" C )  =  if ( ( A  i^i  C )  =  (/) ,  (/) ,  B )
23 incom 3501 . . . . . . . . . . . . . . 15  |-  ( C  i^i  A )  =  ( A  i^i  C
)
24 df-ss 3302 . . . . . . . . . . . . . . . 16  |-  ( C 
C_  A  <->  ( C  i^i  A )  =  C )
2524biimpi 187 . . . . . . . . . . . . . . 15  |-  ( C 
C_  A  ->  ( C  i^i  A )  =  C )
2623, 25syl5eqr 2458 . . . . . . . . . . . . . 14  |-  ( C 
C_  A  ->  ( A  i^i  C )  =  C )
2726adantl 453 . . . . . . . . . . . . 13  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  ( A  i^i  C )  =  C )
28 simpl 444 . . . . . . . . . . . . 13  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  C  =/=  (/) )
2927, 28eqnetrd 2593 . . . . . . . . . . . 12  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  ( A  i^i  C )  =/=  (/) )
30 df-ne 2577 . . . . . . . . . . . . 13  |-  ( ( A  i^i  C )  =/=  (/)  <->  -.  ( A  i^i  C )  =  (/) )
3130biimpi 187 . . . . . . . . . . . 12  |-  ( ( A  i^i  C )  =/=  (/)  ->  -.  ( A  i^i  C )  =  (/) )
32 iffalse 3714 . . . . . . . . . . . 12  |-  ( -.  ( A  i^i  C
)  =  (/)  ->  if ( ( A  i^i  C )  =  (/) ,  (/) ,  B )  =  B )
3329, 31, 323syl 19 . . . . . . . . . . 11  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  if ( ( A  i^i  C )  =  (/) ,  (/) ,  B )  =  B )
3422, 33syl5eq 2456 . . . . . . . . . 10  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  (
( A  X.  B
) " C )  =  B )
3534difeq1d 3432 . . . . . . . . 9  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  (
( ( A  X.  B ) " C
)  \  B )  =  ( B  \  B ) )
36 difid 3664 . . . . . . . . 9  |-  ( B 
\  B )  =  (/)
3735, 36syl6eq 2460 . . . . . . . 8  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  (
( ( A  X.  B ) " C
)  \  B )  =  (/) )
3821, 37syl5sseq 3364 . . . . . . 7  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  (
( ( R  i^i  ( A  X.  B
) ) " C
)  \  B )  C_  (/) )
39 ss0 3626 . . . . . . 7  |-  ( ( ( ( R  i^i  ( A  X.  B
) ) " C
)  \  B )  C_  (/)  ->  ( ( ( R  i^i  ( A  X.  B ) )
" C )  \  B )  =  (/) )
4038, 39syl 16 . . . . . 6  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  (
( ( R  i^i  ( A  X.  B
) ) " C
)  \  B )  =  (/) )
41 df-ima 4858 . . . . . . . . . . 11  |-  ( ( R  \  ( A  X.  B ) )
" C )  =  ran  ( ( R 
\  ( A  X.  B ) )  |`  C )
42 df-res 4857 . . . . . . . . . . . 12  |-  ( ( R  \  ( A  X.  B ) )  |`  C )  =  ( ( R  \  ( A  X.  B ) )  i^i  ( C  X.  _V ) )
4342rneqi 5063 . . . . . . . . . . 11  |-  ran  (
( R  \  ( A  X.  B ) )  |`  C )  =  ran  ( ( R  \ 
( A  X.  B
) )  i^i  ( C  X.  _V ) )
4441, 43eqtri 2432 . . . . . . . . . 10  |-  ( ( R  \  ( A  X.  B ) )
" C )  =  ran  ( ( R 
\  ( A  X.  B ) )  i^i  ( C  X.  _V ) )
4544ineq1i 3506 . . . . . . . . 9  |-  ( ( ( R  \  ( A  X.  B ) )
" C )  i^i 
B )  =  ( ran  ( ( R 
\  ( A  X.  B ) )  i^i  ( C  X.  _V ) )  i^i  B
)
46 xpss1 4951 . . . . . . . . . . . 12  |-  ( C 
C_  A  ->  ( C  X.  _V )  C_  ( A  X.  _V )
)
47 sslin 3535 . . . . . . . . . . . 12  |-  ( ( C  X.  _V )  C_  ( A  X.  _V )  ->  ( ( R 
\  ( A  X.  B ) )  i^i  ( C  X.  _V ) )  C_  (
( R  \  ( A  X.  B ) )  i^i  ( A  X.  _V ) ) )
48 rnss 5065 . . . . . . . . . . . 12  |-  ( ( ( R  \  ( A  X.  B ) )  i^i  ( C  X.  _V ) )  C_  (
( R  \  ( A  X.  B ) )  i^i  ( A  X.  _V ) )  ->  ran  ( ( R  \ 
( A  X.  B
) )  i^i  ( C  X.  _V ) ) 
C_  ran  ( ( R  \  ( A  X.  B ) )  i^i  ( A  X.  _V ) ) )
4946, 47, 483syl 19 . . . . . . . . . . 11  |-  ( C 
C_  A  ->  ran  ( ( R  \ 
( A  X.  B
) )  i^i  ( C  X.  _V ) ) 
C_  ran  ( ( R  \  ( A  X.  B ) )  i^i  ( A  X.  _V ) ) )
5049adantl 453 . . . . . . . . . 10  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  ran  ( ( R  \ 
( A  X.  B
) )  i^i  ( C  X.  _V ) ) 
C_  ran  ( ( R  \  ( A  X.  B ) )  i^i  ( A  X.  _V ) ) )
51 ssn0 3628 . . . . . . . . . . . 12  |-  ( ( C  C_  A  /\  C  =/=  (/) )  ->  A  =/=  (/) )
5251ancoms 440 . . . . . . . . . . 11  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  A  =/=  (/) )
53 inss1 3529 . . . . . . . . . . . . . . . 16  |-  ( ( A  X.  _V )  i^i  R )  C_  ( A  X.  _V )
54 ssdif 3450 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  X.  _V )  i^i  R )  C_  ( A  X.  _V )  ->  ( ( ( A  X.  _V )  i^i 
R )  \  ( A  X.  B ) ) 
C_  ( ( A  X.  _V )  \ 
( A  X.  B
) ) )
5553, 54ax-mp 8 . . . . . . . . . . . . . . 15  |-  ( ( ( A  X.  _V )  i^i  R )  \ 
( A  X.  B
) )  C_  (
( A  X.  _V )  \  ( A  X.  B ) )
56 incom 3501 . . . . . . . . . . . . . . . 16  |-  ( ( A  X.  _V )  i^i  ( R  \  ( A  X.  B ) ) )  =  ( ( R  \  ( A  X.  B ) )  i^i  ( A  X.  _V ) )
57 indif2 3552 . . . . . . . . . . . . . . . 16  |-  ( ( A  X.  _V )  i^i  ( R  \  ( A  X.  B ) ) )  =  ( ( ( A  X.  _V )  i^i  R )  \ 
( A  X.  B
) )
5856, 57eqtr3i 2434 . . . . . . . . . . . . . . 15  |-  ( ( R  \  ( A  X.  B ) )  i^i  ( A  X.  _V ) )  =  ( ( ( A  X.  _V )  i^i  R ) 
\  ( A  X.  B ) )
59 difxp2 6349 . . . . . . . . . . . . . . 15  |-  ( A  X.  ( _V  \  B ) )  =  ( ( A  X.  _V )  \  ( A  X.  B ) )
6055, 58, 593sstr4i 3355 . . . . . . . . . . . . . 14  |-  ( ( R  \  ( A  X.  B ) )  i^i  ( A  X.  _V ) )  C_  ( A  X.  ( _V  \  B ) )
61 rnss 5065 . . . . . . . . . . . . . 14  |-  ( ( ( R  \  ( A  X.  B ) )  i^i  ( A  X.  _V ) )  C_  ( A  X.  ( _V  \  B ) )  ->  ran  ( ( R  \ 
( A  X.  B
) )  i^i  ( A  X.  _V ) ) 
C_  ran  ( A  X.  ( _V  \  B
) ) )
6260, 61mp1i 12 . . . . . . . . . . . . 13  |-  ( A  =/=  (/)  ->  ran  ( ( R  \  ( A  X.  B ) )  i^i  ( A  X.  _V ) )  C_  ran  ( A  X.  ( _V  \  B ) ) )
63 rnxp 5266 . . . . . . . . . . . . 13  |-  ( A  =/=  (/)  ->  ran  ( A  X.  ( _V  \  B ) )  =  ( _V  \  B
) )
6462, 63sseqtrd 3352 . . . . . . . . . . . 12  |-  ( A  =/=  (/)  ->  ran  ( ( R  \  ( A  X.  B ) )  i^i  ( A  X.  _V ) )  C_  ( _V  \  B ) )
65 disj2 3643 . . . . . . . . . . . 12  |-  ( ( ran  ( ( R 
\  ( A  X.  B ) )  i^i  ( A  X.  _V ) )  i^i  B
)  =  (/)  <->  ran  ( ( R  \  ( A  X.  B ) )  i^i  ( A  X.  _V ) )  C_  ( _V  \  B ) )
6664, 65sylibr 204 . . . . . . . . . . 11  |-  ( A  =/=  (/)  ->  ( ran  ( ( R  \ 
( A  X.  B
) )  i^i  ( A  X.  _V ) )  i^i  B )  =  (/) )
6752, 66syl 16 . . . . . . . . . 10  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  ( ran  ( ( R  \ 
( A  X.  B
) )  i^i  ( A  X.  _V ) )  i^i  B )  =  (/) )
68 ssdisj 3645 . . . . . . . . . 10  |-  ( ( ran  ( ( R 
\  ( A  X.  B ) )  i^i  ( C  X.  _V ) )  C_  ran  ( ( R  \ 
( A  X.  B
) )  i^i  ( A  X.  _V ) )  /\  ( ran  (
( R  \  ( A  X.  B ) )  i^i  ( A  X.  _V ) )  i^i  B
)  =  (/) )  -> 
( ran  ( ( R  \  ( A  X.  B ) )  i^i  ( C  X.  _V ) )  i^i  B
)  =  (/) )
6950, 67, 68syl2anc 643 . . . . . . . . 9  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  ( ran  ( ( R  \ 
( A  X.  B
) )  i^i  ( C  X.  _V ) )  i^i  B )  =  (/) )
7045, 69syl5eq 2456 . . . . . . . 8  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  (
( ( R  \ 
( A  X.  B
) ) " C
)  i^i  B )  =  (/) )
71 disj3 3640 . . . . . . . 8  |-  ( ( ( ( R  \ 
( A  X.  B
) ) " C
)  i^i  B )  =  (/)  <->  ( ( R 
\  ( A  X.  B ) ) " C )  =  ( ( ( R  \ 
( A  X.  B
) ) " C
)  \  B )
)
7270, 71sylib 189 . . . . . . 7  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  (
( R  \  ( A  X.  B ) )
" C )  =  ( ( ( R 
\  ( A  X.  B ) ) " C )  \  B
) )
7372eqcomd 2417 . . . . . 6  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  (
( ( R  \ 
( A  X.  B
) ) " C
)  \  B )  =  ( ( R 
\  ( A  X.  B ) ) " C ) )
7440, 73uneq12d 3470 . . . . 5  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  (
( ( ( R  i^i  ( A  X.  B ) ) " C )  \  B
)  u.  ( ( ( R  \  ( A  X.  B ) )
" C )  \  B ) )  =  ( (/)  u.  (
( R  \  ( A  X.  B ) )
" C ) ) )
7517, 74syl5eq 2456 . . . 4  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  (
( R " C
)  \  B )  =  ( (/)  u.  (
( R  \  ( A  X.  B ) )
" C ) ) )
76 uncom 3459 . . . . 5  |-  ( (/)  u.  ( ( R  \ 
( A  X.  B
) ) " C
) )  =  ( ( ( R  \ 
( A  X.  B
) ) " C
)  u.  (/) )
77 un0 3620 . . . . 5  |-  ( ( ( R  \  ( A  X.  B ) )
" C )  u.  (/) )  =  (
( R  \  ( A  X.  B ) )
" C )
7876, 77eqtr2i 2433 . . . 4  |-  ( ( R  \  ( A  X.  B ) )
" C )  =  ( (/)  u.  (
( R  \  ( A  X.  B ) )
" C ) )
7975, 78syl6reqr 2463 . . 3  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  (
( R  \  ( A  X.  B ) )
" C )  =  ( ( R " C )  \  B
) )
8079ancoms 440 . 2  |-  ( ( C  C_  A  /\  C  =/=  (/) )  ->  (
( R  \  ( A  X.  B ) )
" C )  =  ( ( R " C )  \  B
) )
8110, 80pm2.61dane 2653 1  |-  ( C 
C_  A  ->  (
( R  \  ( A  X.  B ) )
" C )  =  ( ( R " C )  \  B
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    =/= wne 2575   _Vcvv 2924    \ cdif 3285    u. cun 3286    i^i cin 3287    C_ wss 3288   (/)c0 3596   ifcif 3707    X. cxp 4843   ran crn 4846    |` cres 4847   "cima 4848
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-br 4181  df-opab 4235  df-xp 4851  df-rel 4852  df-cnv 4853  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858
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